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Magma
magma: G := TransitiveGroup(16, 13);
Group invariants
Abstract group: | $D_{8}$ | magma: IdentifyGroup(G);
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Order: | $16=2^{4}$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $3$ | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $16$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(13,15)(14,16)$, $(1,4,6,8,10,12,14,15)(2,3,5,7,9,11,13,16)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Low degree siblings
8T6 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{16}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{8}$ | $1$ | $2$ | $8$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$ |
2B | $2^{8}$ | $4$ | $2$ | $8$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,16)$ |
2C | $2^{8}$ | $4$ | $2$ | $8$ | $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,15)(14,16)$ |
4A | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$ |
8A1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 4, 6, 8,10,12,14,15)( 2, 3, 5, 7, 9,11,13,16)$ |
8A3 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1,12, 6,15,10, 4,14, 8)( 2,11, 5,16, 9, 3,13, 7)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 4A | 8A1 | 8A3 | ||
Size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 2A | 4A | 4A | |
Type | ||||||||
16.7.1a | R | |||||||
16.7.1b | R | |||||||
16.7.1c | R | |||||||
16.7.1d | R | |||||||
16.7.2a | R | |||||||
16.7.2b1 | R | |||||||
16.7.2b2 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed